Delayed Feedback Control on a Class of Generalized Gyroscope Systems under Parametric Excitation

AbstractThe nonlinear dynamics of the parametrically excited vibrations of a class of generalized gyroscope systems under delayed feedback control is investigated by the averaging method and simulations in this paper. The influence of feedback control on the stability of the trivial solution and the amplitude of the periodic vibrations is presented based on Routh-Hurwitz criterion and the Levenberg-Marquardt method respectively. It is shown that the stability of the trivial solution can be varied when feedback control and time delay are employed. The amplitudes of periodic solutions can also be modulated greatly by feedback gain and time delay. However, the influence of time delay on amplitudes is periodic. The simulations obtained by numerically integrating the original system are in good agreement with the analytical results

On deciding stability of high frequency amplifiers

International audienceHyper-frequency amplifiers are nonlinear because of active components (diodes…) and infinite-dimensional because of transmission lines, here modelled by telegrapher’s PDEs. Using a resolved form for the latter, one obtains delay difference and differential equations, strictly more general than "neutral type equations". CAD tools provide estimates of frequency responses, but nothing on their stability. We discuss the mathematical foundations of deducing stability from various frequency responses. After linearisation around periodic solutions, this involves spectral properties of the monodromy operator of time-periodic delay systems that are far from well known. We will outline these results, mostly relying on an appropriate example

Stationary localized structures and the effect of the delayed feedback in the Brusselator model

The Brusselator reaction-diffusion model is a paradigm for the understanding of dissipative structures in systems out of equilibrium. In the first part of this paper, we investigate the formation of stationary localized structures in the Brusselator model. By using numerical continuation methods in two spatial dimensions, we establish a bifurcation diagram showing the emergence of localized spots. We characterize the transition from a single spot to an extended pattern in the form of squares. In the second part, we incorporate delayed feedback control and show that delayed feedback can induce a spontaneous motion of both localized and periodic dissipative structures. We characterize this motion by estimating the threshold and the velocity of the moving dissipative structures.Comment: 18 pages, 11 figure

Dynamics of a Limit Cycle Oscillator under Time Delayed Linear and Nonlinear Feedbacks

We study the effects of time delayed linear and nonlinear feedbacks on the dynamics of a single Hopf bifurcation oscillator. Our numerical and analytic investigations reveal a host of complex temporal phenomena such as phase slips, frequency suppression, multiple periodic states and chaos. Such phenomena are frequently observed in the collective behavior of a large number of coupled limit cycle oscillators. Our time delayed feedback model offers a simple paradigm for obtaining and investigating these temporal states in a single oscillator.We construct a detailed bifurcation diagram of the oscillator as a function of the time delay parameter and the driving strengths of the feedback terms. We find some new states in the presence of the quadratic nonlinear feedback term with interesting characteristics like birhythmicity, phase reversals, radial trapping, phase jumps and spiraling patterns in the amplitude space. Our results may find useful applications in physical, chemical or biological systems.Comment: VERSION 4: Fig. 10(d) added, an uncited reference removed; (To appear in Physica D) (17 pages, 21 figures, two column, aps RevTeX); VERSION 3: Revised. In Section 2, small tau approximation added; Section 3 is divided into subsections; periodic solution discussed in detail; Figs. 7 and 11 discarded; Figs. 12 and 14 altered; three new figures (now Figs. 10, 11 and 21) added. VERSION 2: Figs. 1 and 2 replace

Almost periodic solutions of retarded SICNNs with functional response on piecewise constant argument

We consider a new model for shunting inhibitory cellular neural networks, retarded functional differential equations with piecewise constant argument. The existence and exponential stability of almost periodic solutions are investigated. An illustrative example is provided.Comment: 24 pages, 1 figur